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Simplified problem: Given a $d$-times nested convolution of an input function $g(x):\mathbb{R}\mapsto \mathbb{R}$ with the same band-limited smooth function $f(x):\mathbb{R}\mapsto \mathbb{R}$. I am interested in the value of this $d$-dimensional integral $I$

$I=<f*<f*<...<f*g>..>$,

which I compute using Monte Carlo (this is a necessary evil for the non-simplified problem).

Question: Given that we can estimate spectrum of both $f$ and $g$, how to construct the best importance function for the MC integration? In other words, how to distribute sampling density to different integration dimensions?

Intuitively, I would probably sample the outermost convolution most often, since its integrand has the largest spectral bandwidth (each convolution is a smoothing of the initial function).

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  • $\begingroup$ Why not work in Fourier space and construct a suitable numerical approximation of the n-fold product $(\hat{f})^n$ over the entire band, then multiply by $\hat{g}$ and perform the inverse transform? $\endgroup$
    – Steve Huntsman
    Commented Oct 2, 2013 at 18:30
  • $\begingroup$ Fair point. I described a simplified problem (just edited it to make it clear). It is not possible to do a Fourier transform for the real problem (the functions themselves are high-dimensional there), so I have to stick to Monte Carlo. $\endgroup$
    – Anton
    Commented Oct 2, 2013 at 23:23

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